Vector Calculus

(a) C: (x,y,z)=(4, t, t^2), t=-1~3
ds=√(0+1+4t^2) dt
∫_C xy ds= ∫_[-1,3] 4t√(1+4t^2) dt
= (1/3)(1+4t^2)^(3/2) sub. t=-1~3
= (37√37 - 5√5)/3

(b)
(i) curl(F)= ( x-x, y-y, z-z) =(0, 0, 0), so that F is irrotational.
div(F)=2x+0+0=2x, not zero, so that F is not solenoidal
(ii) ∂φ/∂x= yz , then φ=xyz+x^3/ 3+ C1(y, z)
∂φ/∂y=xz, then φ=xyz+C2(z,x)
∂φ/∂z=xy, then φ=xyz+C3(x,y)
thus φ=xyz+ x^3 / 3+ C (constant independent on x,y,z)